Minimum entangled state dimension required for pseudo-telepathy

Pseudo-telepathy provides an intuitive way of looking at Bell’s inequalities, in which it is often obvious that feats achievable by use of quantum entanglement would be classically impossible. A two-player pseudo-telepathy game proceeds as follows: Alice and Bob are individually asked a question and they must provide an answer. They are not allowed any form of communication once the questions are asked, but they may have agreed on a common strategy prior to the execution of the game. We say that they win the game if the questions and answers fulfil a specific relation. A game exhibits pseudo-telepathy if there is a quantum strategy that makes Alice and Bob win the game for all possible questions, provided they share prior entanglement, whereas it would be impossible to win this game systematically in a classical setting. In this paper, we show that any two-player pseudo-telepathy game requires the quantum players to share an entangled quantum system of dimension at least 3× 3. This is optimal for twoplayer games, but the most efficient pseudo-telepathy game possible, in terms of total dimension, involves three players who share a quantum system of dimension 2× 2× 2. Supported in part by the Natural Sciences and Engineering Research Council of Canada (Nserc), the Canadian Institute for Advanced Research (Ciar), the Mathematics of Information Technology and Complex Systems Network (Mitacs) and the Canada Research Chair Programme. Supported in part by Nserc, Ciar, Mitacs and Québec’s Fqrnt.